   echo = 1 (on)
? gettime;p2=Pol([1,3021,-786303,-6826636057,-546603588746,3853890514072057]
)
x^5 + 3021*x^4 - 786303*x^3 - 6826636057*x^2 - 546603588746*x + 385389051407
2057
? fa=[11699,6;2392997,2;4987333019653,2]

[11699 6]

[2392997 2]

[4987333019653 2]

? setrand(1);a=matrix(3,5,j,k,vectorv(5,l,random\10^8));
? setrand(1);as=matrix(3,3,j,k,vectorv(5,l,random\10^8));
? nfpol=x^5-5*x^3+5*x+25;nf=nfinit(nfpol)
[x^5 - 5*x^3 + 5*x + 25, [1, 2], 595125, 45, [[1, -1.08911514572050482502495
27946671612684, -2.4285174907194186068992069565359418365, 0.7194669112891317
8943997506477288225737, -2.5558200350691694950646071159426779972; 1, -0.1383
8372073406036365047976417441696637 - 0.4918163765776864349975328551474152510
7*I, 1.9647119211288133163138753392090569931 + 0.809714924188978951282940822
19556466857*I, -0.072312766896812300380582649294307897075 + 2.19808037538462
76641195195160383234878*I, -0.98796319352507039803950539735452837193 + 1.570
1452385894131769052374806001981109*I; 1, 1.682941293594312776162956161507997
6006 + 2.0500351226010726172974286983598602164*I, -0.75045317576910401286427
186094108607489 + 1.3101462685358123283560773619310445916*I, -0.787420688747
75359433940488309213323154 + 2.1336633893126618034168454610457936018*I, 1.26
58732110596551455718089553258673705 - 2.716479010374315056657802803578983483
5*I], [1, -1.0891151457205048250249527946671612684, -2.428517490719418606899
2069565359418365, 0.71946691128913178943997506477288225737, -2.5558200350691
694950646071159426779972; 1, -0.63020009731174679864801261932183221743, 2.77
44268453177922675968161614046216617, 2.1257676084878153637389368667440155907
, 0.58218204506434277886573208324566973897; 1, 0.353432655843626071347053090
97299828470, 1.1549969969398343650309345170134923246, -2.2703931422814399645
001021653326313849, -2.5581084321144835749447428779547264828; 1, 3.732976416
1953853934603848598678578170, 0.55969309276670831549180550098995851667, 1.34
62427005649082090774405779536603703, -1.450605799314659911085993848253116112
9; 1, -0.36709382900675984113447253685186261580, -2.060599444304916341220349
2228721306665, -2.9210840780604153977562503441379268334, 3.98235222143397020
22296117589048508540], [1, -1, -2, 1, -3; 1, -1, 3, 2, 1; 1, 0, 1, -2, -3; 1
, 4, 1, 1, -1; 1, 0, -2, -3, 4], [5, 2, 0, -1, -2; 2, -2, -5, -10, 20; 0, -5
, 10, -10, 5; -1, -10, -10, -17, 1; -2, 20, 5, 1, -8], [345, 0, 200, 110, 17
7; 0, 345, 95, 1, 145; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [63, 3,
 0, -6, -9; 3, 8, -5, -1, 16; 0, -5, 22, -10, 0; -6, -1, -10, -14, -9; -9, 1
6, 0, -9, -2], [345, [138, 117, 330, 288, -636; -172, -88, 65, 118, -116; 53
, 1, 138, -173, 65; 1, -172, 54, 191, 106; 0, 118, 173, 225, -34]]], [-2.428
5174907194186068992069565359418365, 1.9647119211288133163138753392090569931 
+ 0.80971492418897895128294082219556466857*I, -0.750453175769104012864271860
94108607489 + 1.3101462685358123283560773619310445916*I], [1, 1/15*x^4 - 2/3
*x^2 + 1/3*x + 4/3, x, 2/15*x^4 - 1/3*x^2 + 2/3*x - 1/3, -1/15*x^4 + 1/3*x^3
 + 1/3*x^2 - 4/3*x - 2/3], [1, 0, 3, 1, 10; 0, 0, -2, 1, -5; 0, 1, 0, 3, -5;
 0, 0, 1, 1, 10; 0, 0, 0, 3, 0], [1, 0, 0, 0, 0, 0, -1, -1, -2, 4, 0, -1, 3,
 -1, 1, 0, -2, -1, -3, -1, 0, 4, 1, -1, -1; 0, 1, 0, 0, 0, 1, 1, -1, -1, 1, 
0, -1, -2, -1, 1, 0, -1, -1, -1, 3, 0, 1, 1, 3, -3; 0, 0, 1, 0, 0, 0, 0, 0, 
1, -1, 1, 0, 0, 0, -2, 0, 1, 0, -1, -1, 0, -1, -2, -1, -1; 0, 0, 0, 1, 0, 0,
 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 0, -1; 0, 0, 0, 0, 1, 0,
 -1, -1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1, 0, 0, -1]]
? nfinit(nfpol,2)
[x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.089115145
7205048250249527946671612684, -2.4285174907194186068992069565359418365, 0.71
946691128913178943997506477288225735, -2.55582003506916949506460711594267799
71; 1, -0.13838372073406036365047976417441696637 + 0.49181637657768643499753
285514741525107*I, 1.9647119211288133163138753392090569931 - 0.8097149241889
7895128294082219556466857*I, -0.072312766896812300380582649294307897123 - 2.
1980803753846276641195195160383234878*I, -0.98796319352507039803950539735452
837196 - 1.5701452385894131769052374806001981109*I; 1, 1.6829412935943127761
629561615079976006 + 2.0500351226010726172974286983598602164*I, -0.750453175
76910401286427186094108607490 + 1.3101462685358123283560773619310445915*I, -
0.78742068874775359433940488309213323160 + 2.1336633893126618034168454610457
936016*I, 1.2658732110596551455718089553258673705 - 2.7164790103743150566578
028035789834836*I], [1, -1.0891151457205048250249527946671612684, -2.4285174
907194186068992069565359418365, 0.71946691128913178943997506477288225735, -2
.5558200350691694950646071159426779971; 1, 0.3534326558436260713470530909729
9828470, 1.1549969969398343650309345170134923246, -2.27039314228143996450010
21653326313849, -2.5581084321144835749447428779547264828; 1, -0.630200097311
74679864801261932183221744, 2.7744268453177922675968161614046216617, 2.12576
76084878153637389368667440155906, 0.58218204506434277886573208324566973893; 
1, 3.7329764161953853934603848598678578170, 0.559693092766708315491805500989
95851657, 1.3462427005649082090774405779536603700, -1.4506057993146599110859
938482531161132; 1, -0.36709382900675984113447253685186261580, -2.0605994443
049163412203492228721306664, -2.9210840780604153977562503441379268332, 3.982
3522214339702022296117589048508541], [1, -1, -2, 1, -3; 1, 0, 1, -2, -3; 1, 
-1, 3, 2, 1; 1, 4, 1, 1, -1; 1, 0, -2, -3, 4], [5, 2, 0, -1, -2; 2, -2, -5, 
-10, 20; 0, -5, 10, -10, 5; -1, -10, -10, -17, 1; -2, 20, 5, 1, -8], [345, 0
, 200, 110, 177; 0, 345, 95, 1, 145; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0, 
0, 1], [63, 3, 0, -6, -9; 3, 8, -5, -1, 16; 0, -5, 22, -10, 0; -6, -1, -10, 
-14, -9; -9, 16, 0, -9, -2], [345, [138, 117, 330, 288, -636; -172, -88, 65,
 118, -116; 53, 1, 138, -173, 65; 1, -172, 54, 191, 106; 0, 118, 173, 225, -
34]]], [-1.0891151457205048250249527946671612684, -0.13838372073406036365047
976417441696637 + 0.49181637657768643499753285514741525107*I, 1.682941293594
3127761629561615079976006 + 2.0500351226010726172974286983598602164*I], [1, 
x, -1/2*x^4 + 3/2*x^3 - 5/2*x^2 - 2*x + 1, -1/2*x^4 + x^3 - x^2 - 9/2*x - 1,
 -1/2*x^4 + x^3 - 2*x^2 - 7/2*x - 2], [1, 0, -1, -7, -14; 0, 1, 1, -2, -15; 
0, 0, 0, 2, 4; 0, 0, 1, 1, -2; 0, 0, -1, -3, -4], [1, 0, 0, 0, 0, 0, -1, -1,
 -2, 4, 0, -1, 3, -1, 1, 0, -2, -1, -3, -1, 0, 4, 1, -1, -1; 0, 1, 0, 0, 0, 
1, 1, -1, -1, 1, 0, -1, -2, -1, 1, 0, -1, -1, -1, 3, 0, 1, 1, 3, -3; 0, 0, 1
, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, -2, 0, 1, 0, -1, -1, 0, -1, -2, -1, -1; 
0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 0, -1; 
0, 0, 0, 0, 1, 0, -1, -1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1, 0, 0,
 -1]]
? nfinit(nfpol,3)
[[x^5 - 2*x^4 + 3*x^3 + 8*x^2 + 3*x + 2, [1, 2], 595125, 4, [[1, -1.08911514
57205048250249527946671612684, -2.4285174907194186068992069565359418365, 0.7
1946691128913178943997506477288225735, -2.5558200350691694950646071159426779
971; 1, -0.13838372073406036365047976417441696637 + 0.4918163765776864349975
3285514741525107*I, 1.9647119211288133163138753392090569931 - 0.809714924188
97895128294082219556466857*I, -0.072312766896812300380582649294307897123 - 2
.1980803753846276641195195160383234878*I, -0.9879631935250703980395053973545
2837196 - 1.5701452385894131769052374806001981109*I; 1, 1.682941293594312776
1629561615079976006 + 2.0500351226010726172974286983598602164*I, -0.75045317
576910401286427186094108607490 + 1.3101462685358123283560773619310445915*I, 
-0.78742068874775359433940488309213323160 + 2.133663389312661803416845461045
7936016*I, 1.2658732110596551455718089553258673705 - 2.716479010374315056657
8028035789834836*I], [1, -1.0891151457205048250249527946671612684, -2.428517
4907194186068992069565359418365, 0.71946691128913178943997506477288225735, -
2.5558200350691694950646071159426779971; 1, 0.353432655843626071347053090972
99828470, 1.1549969969398343650309345170134923246, -2.2703931422814399645001
021653326313849, -2.5581084321144835749447428779547264828; 1, -0.63020009731
174679864801261932183221744, 2.7744268453177922675968161614046216617, 2.1257
676084878153637389368667440155906, 0.58218204506434277886573208324566973893;
 1, 3.7329764161953853934603848598678578170, 0.55969309276670831549180550098
995851657, 1.3462427005649082090774405779536603700, -1.450605799314659911085
9938482531161132; 1, -0.36709382900675984113447253685186261580, -2.060599444
3049163412203492228721306664, -2.9210840780604153977562503441379268332, 3.98
23522214339702022296117589048508541], [1, -1, -2, 1, -3; 1, 0, 1, -2, -3; 1,
 -1, 3, 2, 1; 1, 4, 1, 1, -1; 1, 0, -2, -3, 4], [5, 2, 0, -1, -2; 2, -2, -5,
 -10, 20; 0, -5, 10, -10, 5; -1, -10, -10, -17, 1; -2, 20, 5, 1, -8], [345, 
0, 200, 110, 177; 0, 345, 95, 1, 145; 0, 0, 5, 4, 3; 0, 0, 0, 1, 0; 0, 0, 0,
 0, 1], [63, 3, 0, -6, -9; 3, 8, -5, -1, 16; 0, -5, 22, -10, 0; -6, -1, -10,
 -14, -9; -9, 16, 0, -9, -2], [345, [138, 117, 330, 288, -636; -172, -88, 65
, 118, -116; 53, 1, 138, -173, 65; 1, -172, 54, 191, 106; 0, 118, 173, 225, 
-34]]], [-1.0891151457205048250249527946671612684, -0.1383837207340603636504
7976417441696637 + 0.49181637657768643499753285514741525107*I, 1.68294129359
43127761629561615079976006 + 2.0500351226010726172974286983598602164*I], [1,
 x, -1/2*x^4 + 3/2*x^3 - 5/2*x^2 - 2*x + 1, -1/2*x^4 + x^3 - x^2 - 9/2*x - 1
, -1/2*x^4 + x^3 - 2*x^2 - 7/2*x - 2], [1, 0, -1, -7, -14; 0, 1, 1, -2, -15;
 0, 0, 0, 2, 4; 0, 0, 1, 1, -2; 0, 0, -1, -3, -4], [1, 0, 0, 0, 0, 0, -1, -1
, -2, 4, 0, -1, 3, -1, 1, 0, -2, -1, -3, -1, 0, 4, 1, -1, -1; 0, 1, 0, 0, 0,
 1, 1, -1, -1, 1, 0, -1, -2, -1, 1, 0, -1, -1, -1, 3, 0, 1, 1, 3, -3; 0, 0, 
1, 0, 0, 0, 0, 0, 1, -1, 1, 0, 0, 0, -2, 0, 1, 0, -1, -1, 0, -1, -2, -1, -1;
 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 2, 1, 0, 1, 0, 0, 0, 0, 2, 0, -1;
 0, 0, 0, 0, 1, 0, -1, -1, -1, 1, 0, -1, 0, 1, 0, 0, -1, 1, 0, 0, 1, 1, 0, 0
, -1]], Mod(-1/2*x^4 + 3/2*x^3 - 5/2*x^2 - 2*x + 1, x^5 - 2*x^4 + 3*x^3 + 8*
x^2 + 3*x + 2)]
? nf3=nfinit(x^6+108);
? nf4=nfinit(x^3-10*x+8)
[x^3 - 10*x + 8, [3, 0], 568, 2, [[1, -0.36332823793268357037416860931988791
957, -3.1413361156553641347759399165844441384; 1, -1.76155718183189058754537
11274124874988, 2.6261980685272936133764995500786243868; 1, 3.12488541976457
41579195397367323754184, -0.48486195287192947860055963349418024847], [1, -0.
36332823793268357037416860931988791957, -3.141336115655364134775939916584444
1384; 1, -1.7615571818318905875453711274124874988, 2.62619806852729361337649
95500786243868; 1, 3.1248854197645741579195397367323754184, -0.4848619528719
2947860055963349418024847], [1, 0, -3; 1, -2, 3; 1, 3, 0], [3, 1, -1; 1, 13,
 -5; -1, -5, 17], [284, 76, 46; 0, 2, 0; 0, 0, 1], [98, -6, 4; -6, 25, 7; 4,
 7, 19], [284, [60, 418, -204; 105, 270, -2; 1, 104, -46]]], [-3.50466435358
80477051501085259043320579, 0.86464088669540302583112842266613688801, 2.6400
234668926446793189801032381951699], [1, 1/2*x^2 + x - 3, -1/2*x^2 + 3], [1, 
0, 6; 0, 1, 0; 0, 1, -2], [1, 0, 0, 0, 4, -2, 0, -2, 6; 0, 1, 0, 1, 2, 0, 0,
 0, -2; 0, 0, 1, 0, 1, -1, 1, -1, -1]]
? setrand(1);bnf2=bnfinit(y^3-y-1);nf2=bnf2[7];
? setrand(1);bnf=bnfinit(x^2-x-57,,[0.2,0.2])
[Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 1, 2]), [-2.71246530518434397468087951060
61300699 - 3.1415926535897932384626433832795028842*I; 2.71246530518434397468
08795106061300699], [-0.92212354848661459835166758997591019381 + 3.141592653
5897932384626433832795028842*I, -1.4227033521190704721778709033666269682 + 3
.1415926535897932384626433832795028842*I, -2.0109798024989157562122634098917
610612 + 3.1415926535897932384626433832795028842*I, 0.E-38, -2.2118855015518
881008546761972154132955, -4.3360743563563867915011632012964092724 + 3.14159
26535897932384626433832795028842*I, -0.9883018503693603305370360822990743872
4, -0.34328764427702709438988786673341921876 + 3.141592653589793238462643383
2795028842*I, 2.7786436070670897068662480029292942634 + 3.141592653589793238
4626433832795028842*I, -0.57883590420958750396177972324249097504; 0.92212354
848661459835166758997591019381, 1.4227033521190704721778709033666269682 + 3.
1415926535897932384626433832795028842*I, 2.010979802498915756212263409891761
0612, 0.E-38, 2.2118855015518881008546761972154132955, 4.3360743563563867915
011632012964092724 + 3.1415926535897932384626433832795028842*I, 0.9883018503
6936033053703608229907438724, 0.34328764427702709438988786673341921877, -2.7
786436070670897068662480029292942633 + 3.14159265358979323846264338327950288
42*I, 0.57883590420958750396177972324249097504], [[3, [-1, 1]~, 1, 1, [0, 57
; 1, 1]], [5, [-2, 1]~, 1, 1, [1, 57; 1, 2]], [11, [-2, 1]~, 1, 1, [1, 57; 1
, 2]], [3, [0, 1]~, 1, 1, [-1, 57; 1, 0]], [5, [1, 1]~, 1, 1, [-2, 57; 1, -1
]], [11, [1, 1]~, 1, 1, [-2, 57; 1, -1]], [19, [0, 1]~, 1, 1, [-1, 57; 1, 0]
], [17, [2, 1]~, 1, 1, [-3, 57; 1, -2]], [19, [-1, 1]~, 1, 1, [0, 57; 1, 1]]
, [17, [-3, 1]~, 1, 1, [2, 57; 1, 3]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [[
1, -7.0663729752107779635959310246705326059; 1, 8.06637297521077796359593102
46705326059], [1, -7.0663729752107779635959310246705326059; 1, 8.06637297521
07779635959310246705326059], [1, -7; 1, 8], [2, 1; 1, 115], [229, 114; 0, 1]
, [115, -1; -1, 2], [229, [114, 57; 1, 115]]], [-7.0663729752107779635959310
246705326059, 8.0663729752107779635959310246705326059], [1, x], [1, 0; 0, 1]
, [1, 0, 0, 57; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.71246530518434397
46808795106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[-0.92212354
848661459835166758997591019381 + 3.1415926535897932384626433832795028842*I, 
0.92212354848661459835166758997591019381]]], 0]
? setrand(1);bnfinit(x^2-x-100000,1)
[Mat(5), Mat([3, 2, 1, 2, 0, 3, 3, 2, 2, 0, 0, 4, 1, 2, 3, 3, 2, 3]), [-129.
82045011403975460991182396195022419 + 4.984517768487935075 E-113*I; 129.8204
5011403975460991182396195022419 - 5.075883674631298447 E-115*I], [-41.811264
589129943393339502258694361489 + 1.3603368248011879837 E-113*I, 9.2399004147
902289816376260438840931575 + 3.1415926535897932384626433832795028842*I, -11
.874609881075406725097315997431161032 + 3.1415926535897932384626433832795028
842*I, 129.82045011403975460991182396195022419 + 6.2831853071795864769252867
665590057684*I, 468.23063545223884606467016754522244212 + 3.1415926535897932
384626433832795028842*I, 194.73067517105963191486773594292533629 + 3.1415926
535897932384626433832795028842*I, -29.936654708054536668242186261263200457 +
 3.1415926535897932384626433832795028842*I, -424.534863752953519162294738193
49039595 + 3.1415926535897932384626433832795028842*I, -255.76460658160162671
307516544486494077 + 3.167351412969930231 E-113*I, -6.7377511782956880607802
359510546381088 + 3.1415926535897932384626433832795028842*I, -471.6134813845
9078444231437736509320889 + 5.684989715587054261 E-113*I, -956.4114698698465
1626671568621635925722 + 9.420840100115689917 E-113*I, -853.8647466999659526
2456074685797842371 + 7.471700769057271314 E-113*I, -356.3305689155217822579
8517105826795550 + 3.329779690558131781 E-113*I, 102.98937362955842429020308
089254908188 + 3.1415926535897932384626433832795028842*I, -144.8006382186883
6768354258315124344337 + 3.1415926535897932384626433832795028842*I, -22.1045
15522613877880850594423816214545 + 3.1415926535897932384626433832795028842*I
, 110.11370104752368909742291612707207725 + 3.141592653589793238462643383279
5028842*I, 343.77379210651143792964748714812080347 + 6.283185307179586476925
2867665590057684*I; 41.811264589129943393339502258694361489, -9.239900414790
2289816376260438840931575, 11.874609881075406725097315997431161032, -129.820
45011403975460991182396195022419 + 3.248565551764031006 E-114*I, -468.230635
45223884606467016754522244212 + 6.2831853071795864769252867665590057684*I, -
194.73067517105963191486773594292533629 + 6.497131103528062012 E-114*I, 29.9
36654708054536668242186261263200457 + 6.283185307179586476925286766559005768
4*I, 424.53486375295351916229473819349039595 - 1.2994262207056124024 E-113*I
, 255.76460658160162671307516544486494077, 6.7377511782956880607802359510546
381088 - 2.030353469852519379 E-115*I, 471.613481384590784442314377365093208
89 + 6.2831853071795864769252867665590057684*I, 956.411469869846516266715686
21635925722 + 6.2831853071795864769252867665590057684*I, 853.864746699965952
62456074685797842371 + 6.2831853071795864769252867665590057684*I, 356.330568
91552178225798517105826795550 + 6.2831853071795864769252867665590057684*I, -
102.98937362955842429020308089254908188 + 3.248565551764031006 E-114*I, 144.
80063821868836768354258315124344337 + 6.283185307179586476925286766559005768
4*I, 22.104515522613877880850594423816214545 - 4.060706939705038757 E-115*I,
 -110.11370104752368909742291612707207725, -343.7737921065114379296474871481
2080347 + 6.497131103528062012 E-114*I], [[2, [1, 1]~, 1, 1, [0, 100000; 1, 
1]], [5, [4, 1]~, 1, 1, [0, 100000; 1, 1]], [13, [-6, 1]~, 1, 1, [5, 100000;
 1, 6]], [2, [2, 1]~, 1, 1, [1, 100000; 1, 2]], [5, [5, 1]~, 1, 1, [-1, 1000
00; 1, 0]], [7, [3, 1]~, 2, 1, [3, 100000; 1, 4]], [13, [5, 1]~, 1, 1, [-6, 
100000; 1, -5]], [17, [14, 1]~, 1, 1, [2, 100000; 1, 3]], [43, [-16, 1]~, 1,
 1, [15, 100000; 1, 16]], [17, [19, 1]~, 1, 1, [-3, 100000; 1, -2]], [23, [-
7, 1]~, 1, 1, [6, 100000; 1, 7]], [23, [6, 1]~, 1, 1, [-7, 100000; 1, -6]], 
[29, [-14, 1]~, 1, 1, [13, 100000; 1, 14]], [29, [13, 1]~, 1, 1, [-14, 10000
0; 1, -13]], [31, [23, 1]~, 1, 1, [7, 100000; 1, 8]], [31, [38, 1]~, 1, 1, [
-8, 100000; 1, -7]], [41, [-7, 1]~, 1, 1, [6, 100000; 1, 7]], [41, [6, 1]~, 
1, 1, [-7, 100000; 1, -6]], [43, [15, 1]~, 1, 1, [-16, 100000; 1, -15]]], 0,
 [x^2 - x - 100000, [2, 0], 400001, 1, [[1, -315.728161301298401613920894896
03747004; 1, 316.72816130129840161392089489603747004], [1, -315.728161301298
40161392089489603747004; 1, 316.72816130129840161392089489603747004], [1, -3
16; 1, 317], [2, 1; 1, 200001], [400001, 200000; 0, 1], [200001, -1; -1, 2],
 [400001, [200000, 100000; 1, 200001]]], [-315.72816130129840161392089489603
747004, 316.72816130129840161392089489603747004], [1, x], [1, 0; 0, 1], [1, 
0, 0, 100000; 0, 1, 1, 1]], [[5, [5], [[2, 1; 0, 1]]], 129.82045011403975460
991182396195022419, 1, [2, -1], [3795548840190137810063032548963691540683360
82609238336*x + 119836165644250789990462835950022871665178127611316131167]],
 [Mat(1), [[0, 0]], [[-41.811264589129943393339502258694361489 + 1.360336824
8011879837 E-113*I, 41.811264589129943393339502258694361489]]], 0]
? \p19
   realprecision = 19 significant digits
? setrand(1);sbnf=bnfcompress(bnfinit(x^3-x^2-14*x-1))
[x^3 - x^2 - 14*x - 1, 3, 10889, [1, x, x^2 - x - 9], [-3.233732695981516673
, -0.07182350902743636345, 4.305556205008953036], 0, Mat(2), Mat([1, 1, 1, 0
, 1, 0, 1, 1]), [9, 15, 16, 39, 17, 10, 33, 57, 69], [2, -1], [[0, 1, 0]~, [
5, 3, 1]~], [[[4, -1, 0]~, [-1, 1, 0]~, [2, 1, 0]~, [10, 5, 1]~, [3, 1, 0]~,
 [3, 0, 0]~, [1, 1, 0]~, [-7, -2, 0]~, [-2, -1, 1]~], 0]]
? \p38
   realprecision = 38 significant digits
? bnr=bnrinit(bnf,[[5,3;0,1],[1,0]],1)
[[Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 1, 2]), [-2.7124653051843439746808795106
061300699 - 3.1415926535897932384626433832795028842*I; 2.7124653051843439746
808795106061300699], [-0.92212354848661459835166758997591019381 + 3.14159265
35897932384626433832795028842*I, -1.4227033521190704721778709033666269682 + 
3.1415926535897932384626433832795028842*I, -2.010979802498915756212263409891
7610612 + 3.1415926535897932384626433832795028842*I, 0.E-38, -2.211885501551
8881008546761972154132955, -4.3360743563563867915011632012964092724 + 3.1415
926535897932384626433832795028842*I, -0.988301850369360330537036082299074387
24, -0.34328764427702709438988786673341921876 + 3.14159265358979323846264338
32795028842*I, 2.7786436070670897068662480029292942634 + 3.14159265358979323
84626433832795028842*I, -0.57883590420958750396177972324249097504; 0.9221235
4848661459835166758997591019381, 1.4227033521190704721778709033666269682 + 3
.1415926535897932384626433832795028842*I, 2.01097980249891575621226340989176
10612, 0.E-38, 2.2118855015518881008546761972154132955, 4.336074356356386791
5011632012964092724 + 3.1415926535897932384626433832795028842*I, 0.988301850
36936033053703608229907438724, 0.34328764427702709438988786673341921877, -2.
7786436070670897068662480029292942633 + 3.1415926535897932384626433832795028
842*I, 0.57883590420958750396177972324249097504], [[3, [-1, 1]~, 1, 1, [0, 5
7; 1, 1]], [5, [-2, 1]~, 1, 1, [1, 57; 1, 2]], [11, [-2, 1]~, 1, 1, [1, 57; 
1, 2]], [3, [0, 1]~, 1, 1, [-1, 57; 1, 0]], [5, [1, 1]~, 1, 1, [-2, 57; 1, -
1]], [11, [1, 1]~, 1, 1, [-2, 57; 1, -1]], [19, [0, 1]~, 1, 1, [-1, 57; 1, 0
]], [17, [2, 1]~, 1, 1, [-3, 57; 1, -2]], [19, [-1, 1]~, 1, 1, [0, 57; 1, 1]
], [17, [-3, 1]~, 1, 1, [2, 57; 1, 3]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [
[1, -7.0663729752107779635959310246705326059; 1, 8.0663729752107779635959310
246705326059], [1, -7.0663729752107779635959310246705326059; 1, 8.0663729752
107779635959310246705326059], [1, -7; 1, 8], [2, 1; 1, 115], [229, 114; 0, 1
], [115, -1; -1, 2], [229, [114, 57; 1, 115]]], [-7.066372975210777963595931
0246705326059, 8.0663729752107779635959310246705326059], [1, x], [1, 0; 0, 1
], [1, 0, 0, 57; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.7124653051843439
746808795106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[-0.9221235
4848661459835166758997591019381 + 3.1415926535897932384626433832795028842*I,
 0.92212354848661459835166758997591019381]]], [0, [Mat([[5, 1]~, 1])]]], [[[
5, 3; 0, 1], [1, 0]], [8, [4, 2], [2, [-4, 0]~]], Mat([[5, [-2, 1]~, 1, 1, [
1, 1]~], 1]), [[[[4], [[2, 0]~], [[2, 0]~], [Vecsmall([0])], 1]], [[2], [-4]
, [Vecsmall([1])]]], [1, 0; 0, 1]], [1], Mat([1, -3, -6]), [12, [12], [[3, 2
; 0, 1]]], [[0, 1; 0, 0], [-1, -1; 1, -1], 1]]
? rnfinit(nf2,x^5-x-2)
[x^5 - x - 2, [], [[49744, 0, 0; 0, 49744, 0; 0, 0, 49744], 3109], 1, [], []
, [[1, x, x^2, x^3, x^4], [1, 1, 1, 1, 1]], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0
, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [], [y^3 - y - 1, [1, 1], -23, 
1, [[1, 0.75487766624669276004950889635852869190, 1.324717957244746025960908
8544780973407; 1, -0.87743883312334638002475444817926434595 - 0.744861766619
74423659317042860439236724*I, -0.66235897862237301298045442723904867037 + 0.
56227951206230124389918214490937306150*I], [1, 0.754877666246692760049508896
35852869190, 1.3247179572447460259609088544780973407; 1, -1.6223005997430906
166179248767836567132, -0.10007946656007176908127228232967560887; 1, -0.1325
7706650360214343158401957487197871, -1.2246384906846742568796365721484217319
], [1, 1, 1; 1, -2, 0; 1, 0, -1], [3, -1, 0; -1, 1, 3; 0, 3, 2], [23, 16, 13
; 0, 1, 0; 0, 0, 1], [7, -2, 3; -2, -6, 9; 3, 9, -2], [23, [10, 1, 8; 7, 3, 
1; 1, 7, 10]]], [1.3247179572447460259609088544780973407, -0.662358978622373
01298045442723904867037 + 0.56227951206230124389918214490937306150*I], [1, y
^2 - 1, y], [1, 0, 1; 0, 0, 1; 0, 1, 0], [1, 0, 0, 0, 0, 1, 0, 1, 1; 0, 1, 0
, 1, -1, 0, 0, 0, 1; 0, 0, 1, 0, 1, 0, 1, 0, 0]], [x^15 - 5*x^13 + 5*x^12 + 
7*x^11 - 26*x^10 - 5*x^9 + 45*x^8 + 158*x^7 - 98*x^6 + 110*x^5 - 190*x^4 + 1
89*x^3 + 144*x^2 + 25*x + 1, Mod(39516536165538345/83718587879473471*x^14 - 
6500512476832995/83718587879473471*x^13 - 196215472046117185/837185878794734
71*x^12 + 229902227480108910/83718587879473471*x^11 + 237380704030959181/837
18587879473471*x^10 - 1064931988160773805/83718587879473471*x^9 - 2065708667
1714300/83718587879473471*x^8 + 1772885205999206010/83718587879473471*x^7 + 
5952033217241102348/83718587879473471*x^6 - 4838840187320655696/837185878794
73471*x^5 + 5180390720553188700/83718587879473471*x^4 - 8374015687535120430/
83718587879473471*x^3 + 8907744727915040221/83718587879473471*x^2 + 41559766
64123434381/83718587879473471*x + 318920215718580450/83718587879473471, x^15
 - 5*x^13 + 5*x^12 + 7*x^11 - 26*x^10 - 5*x^9 + 45*x^8 + 158*x^7 - 98*x^6 + 
110*x^5 - 190*x^4 + 189*x^3 + 144*x^2 + 25*x + 1), -1], 0]
? bnfcertify(bnf)
1
? setrand(1);bnfinit(x^2-x-100000,1).fu
[Mod(379554884019013781006303254896369154068336082609238336*x + 119836165644
250789990462835950022871665178127611316131167, x^2 - x - 100000)]
? setrand(1);bnfinit(x^4+24*x^2+585*x+1791,,[0.1,0.1])
[Mat(4), Mat([2, 1, 3, 2, 3, 1, 2, 2, 3, 1, 1]), [3.794126968821658934140827
4220859400304 - 5.9870527641550058222864239850468422116*I; -3.79412696882165
89341408274220859400304 - 10.768108054990558116181007392443839838*I], [1.931
3880959585148864147377738135342082 + 6.7981947468208297437939486179100402848
*I, -4.7254964052532309580038722462221429331 + 0.221943760381343073850224888
09902763050*I, 1.7632415262334312620 E-38 + 11.89942426985550872317839202241
4809397*I, 3.6002448814154879622437514823150496413 + 1.220492915912005599415
8078950334411249*I, 4.7598210168009163773481963089927071348 + 6.313223794426
4898919213732432001027695*I, 2.1166141872052248127966075549404537820 + 1.716
2445529054963529871671522966089216*I, 3.608900877574949007758957640959020457
3 + 6.6552021291379246930056968999284187478*I, 6.912126277608598850588527875
1755091479 + 5.7761366926340659348731122471196012302*I, 6.401642724814892838
2486921648958451520 + 11.897303447035512683665384765889975465*I, 4.601217563
6174863718426969977436700928 + 2.6940425133983098127476830433219050411*I, 1.
1242975011626874487128681981558041465 + 4.8854125246380711363993129164689293
417*I, 2.1252701833646858583118137135844245979 + 0.2575954112574825527638982
4853295908856*I; -1.9313880959585148864147377738135342082 + 11.4779708990964
68256601614561140503719*I, 4.7254964052532309580038722462221429331 + 3.45022
94655929042941706462042507500265*I, -1.7632415262334312620 E-38 + 11.8994242
69855508723178392022414809397*I, -3.6002448814154879622437514823150496413 + 
1.6344045998670569238566869279328251070*I, -4.759821016800916377348196308992
7071348 + 10.155241886908343331355909573977342627*I, -2.11661418720522481279
66075549404537820 + 12.220801522533890048715343423700955279*I, -3.6089008775
749490077589576409590204573 + 8.6157366353087901700411766974755497327*I, -6.
9121262776085988505885278751755091479 + 11.254398791706394483844200470782381
254*I, -6.4016427248148928382486921648958451520 + 9.322022242091338719668785
7963846775313*I, -4.6012175636174863718426969977436700927 + 10.4510119180340
36492108959440241465872*I, -1.1242975011626874487128681981558041465 + 9.5935
345958016036298016925885008458522*I, -2.125270183364685858311813713584424597
9 + 12.308775203101690401086675284585052448*I], [[7, [-2, 3, 3, -1]~, 1, 1, 
[3, 1, 56, 11; 4, 6, -7, 45; 4, -1, 7, -4; 1, 5, 0, 6]], [19, [-5, -6, -2, 1
]~, 2, 1, [9, 13, 36, -14; -8, -1, 6, 61; 3, -2, 12, 8; 2, 5, -11, -1]], [7,
 [1, -6, -2, 1]~, 1, 1, [3, 1, -12, -35; 0, 0, 35, 21; -2, -3, 1, 0; 3, 1, 2
, 0]], [7, [2, -6, -2, 1]~, 1, 1, [2, -1, -1, 30; -1, 4, -31, -29; 1, 3, 3, 
1; -3, -2, -2, 4]], [3, [-1, 0, 1, 0]~, 4, 1, [0, 0, 18, -3; 2, 1, 5, 20; 1,
 -1, 1, -2; 1, 2, 1, 1]], [19, [-4, -6, -2, 1]~, 1, 1, [8, 7, 20, -76; 0, 0,
 76, 95; -1, -8, 7, 0; 8, 7, 1, 0]], [19, [11, -6, -2, 1]~, 1, 1, [9, -4, -1
33, -9; -9, 4, 0, -124; -9, 4, 0, 9; -4, -13, 0, 4]], [13, [-1, -6, -2, 1]~,
 1, 1, [-3, 9, 19, -28; -5, -11, 23, 53; 1, -3, -2, 5; 3, 4, -6, -11]], [13,
 [0, -6, -2, 1]~, 1, 1, [4, 14, 89, -24; -3, -4, 21, 122; 6, -5, 10, 3; 5, 1
1, -9, -4]], [13, [8, -6, -2, 1]~, 1, 1, [6, 11, 77, 3; -3, 1, -6, 83; 6, -2
, 12, 3; 2, 8, -9, 1]], [13, [9, -6, -2, 1]~, 1, 1, [-6, 2, -9, 35; -5, -7, 
-40, -38; 1, 4, -5, 5; -4, -3, -6, -7]], [7, [7, -2, -1, 3]~, 1, 1, [2, -1, 
32, 3; 4, 5, 1, 27; 2, -1, 4, -4; 1, 3, 2, 5]]], 0, [x^4 + 24*x^2 + 585*x + 
1791, [0, 2], 18981, 3087, [[1, 0.49999999999999999999999999999999999995 - 0
.86602540378443864676372317075293618353*I, -3.093348807947282815574224326153
1931904 - 0.11742462569605115853137757107804136513*I, -2.6483671128563006823
406123980207685802 + 2.6202063376006319767212962440086284031*I; 1, 0.5000000
0000000000000000000000000000000 - 0.86602540378443864676372317075293618347*I
, 3.5933488079472828155742243261531931904 + 0.983450029480489805295100741830
97754868*I, 1.6483671128563006823406123980207685802 - 2.62020633760063197672
12962440086284032*I], [1, -0.36602540378443864676372317075293618358, -3.2107
734336433339741056018972312345555, -0.02816077525566870561931615401214017711
1; 1, 1.3660254037844386467637231707529361835, -2.97592418225123165704284675
50751518252, -5.2685734504569326590619086420293969833; 1, -0.366025403784438
64676372317075293618347, 4.5767988374277726208693250679841707391, -0.9718392
2474433129438068384598785982299; 1, 1.3660254037844386467637231707529361835,
 2.6098987784667930102791235843222156418, 4.26857345045693265906190864202939
69834], [1, 0, -3, 0; 1, 1, -3, -5; 1, 0, 5, -1; 1, 1, 3, 4], [4, 2, 1, -2; 
2, -2, 2, -1; 1, 2, 43, 34; -2, -1, 34, -8], [2109, 363, 1926, 1236; 0, 3, 0
, 2; 0, 0, 3, 1; 0, 0, 0, 1], [317, 360, 17, -52; 360, -700, 6, 23; 17, 6, 1
2, 46; -52, 23, 46, -58], [2109, [-993, -60, 7618, 2957; 642, -352, -2315, 4
600; 581, -1, -412, -642; 1, 582, 61, -352]]], [4.53833050303826494880783625
42856178008 + 8.0512080116993661743663904106823282345*I, -4.5383305030382649
488078362542856178008 + 0.60904602614502029327084129684703360027*I], [1, -10
/1029*x^3 + 13/343*x^2 - 165/343*x - 1135/343, 17/1029*x^3 - 32/1029*x^2 + 1
09/343*x + 2444/343, -26/1029*x^3 + 170/1029*x^2 - 429/343*x - 3294/343], [1
, 4, 15, -480; 0, -6, -39, 42; 0, -2, 0, 99; 0, 1, 15, 9], [1, 0, 0, 0, 0, -
1, 1, 1, 0, 1, 12, 4, 0, 1, 4, -9; 0, 1, 0, 0, 1, 1, 0, -1, 0, 0, -4, 9, 0, 
-1, 9, 13; 0, 0, 1, 0, 0, 0, 0, -1, 1, 0, 1, 0, 0, -1, 0, 0; 0, 0, 0, 1, 0, 
0, 1, 1, 0, 1, -1, 0, 1, 1, 0, -1]], [[4, [4], [[7, 2, 4, 0; 0, 1, 0, 0; 0, 
0, 1, 0; 0, 0, 0, 1]]], 3.7941269688216589341408274220859400304, 1, [6, 10/1
029*x^3 - 13/343*x^2 + 165/343*x + 1478/343], [1/147*x^3 + 1/147*x^2 - 8/49*
x - 9/49]], [Mat(-1), [[1.9459101490553133051053527434431797297 + 3.52184386
02827267539446763336694683720*I, 1.9459101490553133051053527434431797296 + 3
.5218438602827267539446763336694683721*I]], [[5.8522525002627383340066731999
591847108 + 7.2891806943100772719847567167678332032*I, 9.7150286921797681068
361487475862531267 + 2.6094045420344387591770907735373697693*I]]], 0]
? bnrconductor(bnf,[[25,13;0,1],[1,1]])
[[5, 3; 0, 1], [1, 0]]
? bnrconductorofchar(bnr,[2])
[[5, 3; 0, 1], [0, 0]]
? bnfisprincipal(bnf,[5,1;0,1],0)
[1]~
? bnfisprincipal(bnf,[5,1;0,1])
[[1]~, [2, 1/3]~]
? bnfisunit(bnf,Mod(3405*x-27466,x^2-x-57))
[-4, Mod(1, 2)]~
? \p19
   realprecision = 19 significant digits
? bnfinit(sbnf)
[Mat(2), Mat([1, 1, 1, 0, 1, 0, 1, 1]), [1.173637103435061715 + 3.1415926535
89793239*I, -4.562279014988837911 + 3.141592653589793239*I; -2.6335434327389
76050 + 3.141592653589793239*I, 1.420330600779487358 + 3.141592653589793239*
I; 1.459906329303914335, 3.141948414209350544], [1.246346989334819161, 0.540
4006376129469728 + 3.141592653589793239*I, -0.6926391142471042845 + 3.141592
653589793239*I, -0.8305625946607188642 + 3.141592653589793239*I, -1.99005644
5584799713 + 3.141592653589793239*I, 0, 0.004375616572659815434 + 3.14159265
3589793239*I, -1.977791147836553954 + 3.141592653589793239*I, 0.367726201402
7817707; 0.6716827432867392935, -0.8333219883742404172 + 3.14159265358979323
8*I, -0.2461086674077943078, -1.552661549868775854, 0.5379005671092853266, 0
, -0.8738318043071131266, 0.5774919091398324094 + 3.141592653589793238*I, 0.
9729063188316092381 + 3.141592653589793239*I; -1.918029732621558454 + 3.1415
92653589793239*I, 0.2929213507612934446, 0.9387477816548985924, 2.3832241445
29494719, 1.452155878475514387, 0, 0.8694561877344533112, 1.4002992386967215
45 + 3.141592653589793239*I, -1.340632520234391008 + 3.141592653589793239*I]
, [[3, [-1, 1, 0]~, 1, 1, [1, 1, 1]~], [5, [-1, 1, 0]~, 1, 1, [0, 1, 1]~], [
5, [2, 1, 0]~, 1, 1, [1, -2, 1]~], [13, [19, 1, 0]~, 1, 1, [-2, -6, 1]~], [5
, [3, 1, 0]~, 1, 1, [2, 2, 1]~], [3, [10, 1, 1]~, 1, 2, [-1, 1, 0]~], [11, [
1, 1, 0]~, 1, 1, [-3, -1, 1]~], [19, [-6, 1, 0]~, 1, 1, [6, 6, 1]~], [23, [-
10, 1, 0]~, 1, 1, [-7, 10, 1]~]]~, 0, [x^3 - x^2 - 14*x - 1, [3, 0], 10889, 
1, [[1, -3.233732695981516673, 4.690759845041404812; 1, -0.07182350902743636
345, -8.923017874523549404; 1, 4.305556205008953037, 5.232258029482144592], 
[1, -3.233732695981516673, 4.690759845041404812; 1, -0.07182350902743636345,
 -8.923017874523549404; 1, 4.305556205008953037, 5.232258029482144592], [1, 
-3, 5; 1, 0, -9; 1, 4, 5], [3, 1, 1; 1, 29, 8; 1, 8, 129], [10889, 5698, 899
4; 0, 1, 0; 0, 0, 1], [3677, -121, -21; -121, 386, -23; -21, -23, 86], [1088
9, [1899, 46720, 5235; 5191, 7095, 25956; 1, 5191, 1895]]], [-3.233732695981
516673, -0.07182350902743636345, 4.305556205008953037], [1, x, x^2 - x - 9],
 [1, 0, 9; 0, 1, 1; 0, 0, 1], [1, 0, 0, 0, 9, 1, 0, 1, 44; 0, 1, 0, 1, 1, 5,
 0, 5, 1; 0, 0, 1, 0, 1, 0, 1, 0, -4]], [[2, [2], [[3, 2, 0; 0, 1, 0; 0, 0, 
1]]], 10.34800724602768001, 1, [2, -1], [x, x^2 + 2*x - 4]], [Mat(1), [[0, 0
, 0]], [[1.246346989334819161, 0.6716827432867392935, -1.918029732621558454 
+ 3.141592653589793239*I]]], [[[4, -1, 0]~, [-1, 1, 0]~, [2, 1, 0]~, [10, 5,
 1]~, [3, 1, 0]~, [3, 0, 0]~, [1, 1, 0]~, [-7, -2, 0]~, [-2, -1, 1]~], 0]]
? \p38
   realprecision = 38 significant digits
? bnfnarrow(bnf)
[3, [3], [[3, 2; 0, 1]]]
? bnfsignunit(bnf)

[-1]

[1]

? bnr2=bnrinit(bnf,[[25,13;0,1],[1,1]],1)
[[Mat(3), Mat([1, 2, 1, 2, 1, 2, 1, 1, 2]), [-2.7124653051843439746808795106
061300699 - 3.1415926535897932384626433832795028842*I; 2.7124653051843439746
808795106061300699], [-0.92212354848661459835166758997591019381 + 3.14159265
35897932384626433832795028842*I, -1.4227033521190704721778709033666269682 + 
3.1415926535897932384626433832795028842*I, -2.010979802498915756212263409891
7610612 + 3.1415926535897932384626433832795028842*I, 0.E-38, -2.211885501551
8881008546761972154132955, -4.3360743563563867915011632012964092724 + 3.1415
926535897932384626433832795028842*I, -0.988301850369360330537036082299074387
24, -0.34328764427702709438988786673341921876 + 3.14159265358979323846264338
32795028842*I, 2.7786436070670897068662480029292942634 + 3.14159265358979323
84626433832795028842*I, -0.57883590420958750396177972324249097504; 0.9221235
4848661459835166758997591019381, 1.4227033521190704721778709033666269682 + 3
.1415926535897932384626433832795028842*I, 2.01097980249891575621226340989176
10612, 0.E-38, 2.2118855015518881008546761972154132955, 4.336074356356386791
5011632012964092724 + 3.1415926535897932384626433832795028842*I, 0.988301850
36936033053703608229907438724, 0.34328764427702709438988786673341921877, -2.
7786436070670897068662480029292942633 + 3.1415926535897932384626433832795028
842*I, 0.57883590420958750396177972324249097504], [[3, [-1, 1]~, 1, 1, [0, 5
7; 1, 1]], [5, [-2, 1]~, 1, 1, [1, 57; 1, 2]], [11, [-2, 1]~, 1, 1, [1, 57; 
1, 2]], [3, [0, 1]~, 1, 1, [-1, 57; 1, 0]], [5, [1, 1]~, 1, 1, [-2, 57; 1, -
1]], [11, [1, 1]~, 1, 1, [-2, 57; 1, -1]], [19, [0, 1]~, 1, 1, [-1, 57; 1, 0
]], [17, [2, 1]~, 1, 1, [-3, 57; 1, -2]], [19, [-1, 1]~, 1, 1, [0, 57; 1, 1]
], [17, [-3, 1]~, 1, 1, [2, 57; 1, 3]]], 0, [x^2 - x - 57, [2, 0], 229, 1, [
[1, -7.0663729752107779635959310246705326059; 1, 8.0663729752107779635959310
246705326059], [1, -7.0663729752107779635959310246705326059; 1, 8.0663729752
107779635959310246705326059], [1, -7; 1, 8], [2, 1; 1, 115], [229, 114; 0, 1
], [115, -1; -1, 2], [229, [114, 57; 1, 115]]], [-7.066372975210777963595931
0246705326059, 8.0663729752107779635959310246705326059], [1, x], [1, 0; 0, 1
], [1, 0, 0, 57; 0, 1, 1, 1]], [[3, [3], [[3, 2; 0, 1]]], 2.7124653051843439
746808795106061300699, 1, [2, -1], [x + 7]], [Mat(1), [[0, 0]], [[-0.9221235
4848661459835166758997591019381 + 3.1415926535897932384626433832795028842*I,
 0.92212354848661459835166758997591019381]]], [0, [Mat([[5, 1]~, 1])]]], [[[
25, 13; 0, 1], [1, 1]], [80, [20, 2, 2], [2, [-24, 0]~, [2, 2]~]], Mat([[5, 
[-2, 1]~, 1, 1, [1, 1]~], 2]), [[[[4], [[2, 0]~], [[2, 0]~], [Vecsmall([0, 0
])], 1], [[5], [[6, 0]~], [[6, 0]~], [Vecsmall([0, 0])], Mat([1/5, -13/5])]]
, [[2, 2], [-24, [2, 2]~], [Vecsmall([0, 1]), Vecsmall([1, 1])]]], [1, -12, 
0, 0; 0, 0, 1, 0; 0, 0, 0, 1]], [1], Mat([1, -3, -6, -6]), [12, [12], [[3, 2
; 0, 1]]], [[0, 2, 0; -1, 10, 0], [-2, 0; 0, -10], 2]]
? bnrclassno(bnf,[[5,3;0,1],[1,0]])
12
? lu=ideallist(bnf,55,3);
? bnrclassnolist(bnf,lu)
[[3], [], [3, 3], [3], [6, 6], [], [], [], [3, 3, 3], [], [3, 3], [3, 3], []
, [], [12, 6, 6, 12], [3], [3, 3], [], [9, 9], [6, 6], [], [], [], [], [6, 1
2, 6], [], [3, 3, 3, 3], [], [], [], [], [], [3, 6, 6, 3], [], [], [9, 3, 9]
, [6, 6], [], [], [], [], [], [3, 3], [3, 3], [12, 12, 6, 6, 12, 12], [], []
, [6, 6], [9], [], [3, 3, 3, 3], [], [3, 3], [], [6, 12, 12, 6]]
? bnrdisc(bnr,Mat(6))
[12, 12, 18026977100265125]
? bnrdisc(bnr)
[24, 12, 40621487921685401825918161408203125]
? bnrdisc(bnr2,,,2)
0
? bnrdisc(bnr,Mat(6),,1)
[6, 2, [125, 13; 0, 1]]
? bnrdisc(bnr,,,1)
[12, 1, [1953125, 1160888; 0, 1]]
? bnrdisc(bnr2,,,3)
0
? bnrdisclist(bnf,lu)
[[[6, 6, Mat([229, 3])]], [], [[], []], [[]], [[12, 12, [5, 3; 229, 6]], [12
, 12, [5, 3; 229, 6]]], [], [], [], [[], [], []], [], [[], []], [[], []], []
, [], [[24, 24, [3, 6; 5, 9; 229, 12]], [], [], [24, 24, [3, 6; 5, 9; 229, 1
2]]], [[]], [[], []], [], [[18, 18, [19, 6; 229, 9]], [18, 18, [19, 6; 229, 
9]]], [[], []], [], [], [], [], [[], [24, 24, [5, 12; 229, 12]], []], [], [[
], [], [], []], [], [], [], [], [], [[], [12, 12, [3, 3; 11, 3; 229, 6]], [1
2, 12, [3, 3; 11, 3; 229, 6]], []], [], [], [[18, 18, [2, 12; 3, 12; 229, 9]
], [], [18, 18, [2, 12; 3, 12; 229, 9]]], [[12, 12, [37, 3; 229, 6]], [12, 1
2, [37, 3; 229, 6]]], [], [], [], [], [], [[], []], [[], []], [[], [], [], [
], [], []], [], [], [[12, 12, [2, 12; 3, 3; 229, 6]], [12, 12, [2, 12; 3, 3;
 229, 6]]], [[18, 18, [7, 12; 229, 9]]], [], [[], [], [], []], [], [[], []],
 [], [[], [24, 24, [5, 9; 11, 6; 229, 12]], [24, 24, [5, 9; 11, 6; 229, 12]]
, []]]
? bnrdisclist(bnf,20)
[[[[matrix(0,2), [[6, 6, Mat([229, 3])], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]],
 [], [[Mat([12, 1]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [12, 0, [3, 3; 229, 6
]]]], [Mat([13, 1]), [[0, 0, 0], [12, 6, [-1, 1; 3, 3; 229, 6]], [0, 0, 0], 
[0, 0, 0]]]], [[Mat([10, 1]), [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]]
, [[Mat([20, 1]), [[12, 12, [5, 3; 229, 6]], [0, 0, 0], [0, 0, 0], [24, 0, [
5, 9; 229, 12]]]], [Mat([21, 1]), [[12, 12, [5, 3; 229, 6]], [24, 12, [5, 9;
 229, 12]], [0, 0, 0], [0, 0, 0]]]], [], [], [], [[Mat([12, 2]), [[0, 0, 0],
 [0, 0, 0], [0, 0, 0], [0, 0, 0]]], [[12, 1; 13, 1], [[0, 0, 0], [0, 0, 0], 
[12, 6, [-1, 1; 3, 6; 229, 6]], [24, 0, [3, 12; 229, 12]]]], [Mat([13, 2]), 
[[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]], [], [[Mat([44, 1]), [[0, 0, 
0], [12, 6, [-1, 1; 11, 3; 229, 6]], [0, 0, 0], [0, 0, 0]]], [Mat([45, 1]), 
[[0, 0, 0], [0, 0, 0], [0, 0, 0], [12, 0, [11, 3; 229, 6]]]]], [[[10, 1; 12,
 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]], [[10, 1; 13, 1], [[0, 0,
 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]], [], [], [[[12, 1; 20, 1], [[24, 24,
 [3, 6; 5, 9; 229, 12]], [0, 0, 0], [0, 0, 0], [48, 0, [3, 12; 5, 18; 229, 2
4]]]], [[13, 1; 20, 1], [[0, 0, 0], [24, 12, [3, 6; 5, 6; 229, 12]], [24, 12
, [3, 6; 5, 9; 229, 12]], [48, 0, [3, 12; 5, 18; 229, 24]]]], [[12, 1; 21, 1
], [[0, 0, 0], [0, 0, 0], [24, 12, [3, 6; 5, 9; 229, 12]], [48, 0, [3, 12; 5
, 18; 229, 24]]]], [[13, 1; 21, 1], [[24, 24, [3, 6; 5, 9; 229, 12]], [48, 2
4, [3, 12; 5, 18; 229, 24]], [0, 0, 0], [0, 0, 0]]]], [[Mat([10, 2]), [[0, 0
, 0], [12, 6, [-1, 1; 2, 12; 229, 6]], [12, 6, [-1, 1; 2, 12; 229, 6]], [24,
 0, [2, 36; 229, 12]]]]], [[Mat([68, 1]), [[0, 0, 0], [0, 0, 0], [12, 6, [-1
, 1; 17, 3; 229, 6]], [0, 0, 0]]], [Mat([69, 1]), [[0, 0, 0], [0, 0, 0], [12
, 6, [-1, 1; 17, 3; 229, 6]], [0, 0, 0]]]], [], [[Mat([76, 1]), [[18, 18, [1
9, 6; 229, 9]], [0, 0, 0], [0, 0, 0], [36, 0, [19, 15; 229, 18]]]], [Mat([77
, 1]), [[18, 18, [19, 6; 229, 9]], [36, 18, [-1, 1; 19, 15; 229, 18]], [0, 0
, 0], [0, 0, 0]]]], [[[10, 1; 20, 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 
0, 0]]], [[10, 1; 21, 1], [[0, 0, 0], [0, 0, 0], [0, 0, 0], [0, 0, 0]]]]]]
? bnrisprincipal(bnr,idealprimedec(bnf,7)[1])
[[9]~, [32879/6561, 13958/19683]~]
? dirzetak(nf4,30)
[1, 2, 0, 3, 1, 0, 0, 4, 0, 2, 1, 0, 0, 0, 0, 5, 1, 0, 0, 3, 0, 2, 0, 0, 2, 
0, 1, 0, 1, 0]
? factornf(x^3+x^2-2*x-1,t^3+t^2-2*t-1)

[x + Mod(-t, t^3 + t^2 - 2*t - 1) 1]

[x + Mod(-t^2 + 2, t^3 + t^2 - 2*t - 1) 1]

[x + Mod(t^2 + t - 1, t^3 + t^2 - 2*t - 1) 1]

? vp=idealprimedec(nf,3)[1]
[3, [1, 0, 1, 0, 0]~, 1, 1, [1, -1, -1, -1, 0]~]
? idx=idealhnf(nf,vp)

[3 2 1 0 1]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idy=idealred(nf,idx,[1,5,6])

[5 0 0 0 2]

[0 5 0 0 2]

[0 0 5 0 1]

[0 0 0 5 2]

[0 0 0 0 1]

? idx2=idealmul(nf,idx,idx)

[9 5 7 0 4]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idt=idealmul(nf,idx,idx,1)

[2 0 0 0 0]

[0 2 0 0 0]

[0 0 2 0 0]

[0 0 0 2 1]

[0 0 0 0 1]

? idz=idealintersect(nf,idx,idy)

[15 10 5 0 12]

[0 5 0 0 2]

[0 0 5 0 1]

[0 0 0 5 2]

[0 0 0 0 1]

? aid=[idx,idy,idz,1,idx]
[[3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]
, [5, 0, 0, 0, 2; 0, 5, 0, 0, 2; 0, 0, 5, 0, 1; 0, 0, 0, 5, 2; 0, 0, 0, 0, 1
], [15, 10, 5, 0, 12; 0, 5, 0, 0, 2; 0, 0, 5, 0, 1; 0, 0, 0, 5, 2; 0, 0, 0, 
0, 1], 1, [3, 2, 1, 0, 1; 0, 1, 0, 0, 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0,
 0, 0, 1]]
? idealadd(nf,idx,idy)

[1 0 0 0 0]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idealaddtoone(nf,idx,idy)
[[0, -1, -3, -1, 2]~, [1, 1, 3, 1, -2]~]
? idealaddtoone(nf,[idy,idx])
[[-5, 0, 0, 0, 0]~, [6, 0, 0, 0, 0]~]
? idealappr(nf,idy)
[-1, 4, 2, -1, -3]~
? idealappr(nf,idealfactor(nf,idy),1)
[-1, 4, 2, -1, -3]~
? idealcoprime(nf,idx,idx)
[-1/3, 1/3, 1/3, 1/3, 0]~
? idealdiv(nf,idy,idt)

[5 0 5/2 0 1]

[0 5/2 0 0 1]

[0 0 5/2 0 1/2]

[0 0 0 5/2 1]

[0 0 0 0 1/2]

? idealdiv(nf,idx2,idx,1)

[3 2 1 0 1]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idealfactor(nf,idz)

[[3, [1, 0, 1, 0, 0]~, 1, 1, [1, -1, -1, -1, 0]~] 1]

[[5, [-1, 0, 0, 0, 2]~, 4, 1, [2, 2, 1, 2, 1]~] 3]

[[5, [2, 0, 0, 0, -2]~, 1, 1, [2, 0, 3, 0, 1]~] 1]

? idealhnf(nf,vp[2],3)

[3 2 1 0 1]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? ideallist(bnf,20)
[[[1, 0; 0, 1]], [], [[3, 2; 0, 1], [3, 0; 0, 1]], [[2, 0; 0, 2]], [[5, 3; 0
, 1], [5, 1; 0, 1]], [], [], [], [[9, 5; 0, 1], [3, 0; 0, 3], [9, 3; 0, 1]],
 [], [[11, 9; 0, 1], [11, 1; 0, 1]], [[6, 4; 0, 2], [6, 0; 0, 2]], [], [], [
[15, 8; 0, 1], [15, 3; 0, 1], [15, 11; 0, 1], [15, 6; 0, 1]], [[4, 0; 0, 4]]
, [[17, 14; 0, 1], [17, 2; 0, 1]], [], [[19, 18; 0, 1], [19, 0; 0, 1]], [[10
, 6; 0, 2], [10, 2; 0, 2]]]
? bid=idealstar(nf2,54)
[[[54, 0, 0; 0, 54, 0; 0, 0, 54], [0]], [132678, [1638, 9, 9]], [[2, [2, 0, 
0]~, 1, 3, 1], 1; [3, [3, 0, 0]~, 1, 3, 1], 3], [[[[7], [[2, 1, 0]~], [[-26,
 -27, 0]~], [Vecsmall([])], 1]], [[[26], [[2, 0, 2]~], [[-25, 0, 2]~], [Vecs
mall([])], 1], [[3, 3, 3], [[4, 0, 0]~, [1, 3, 0]~, [1, 0, 3]~], [[-23, 0, 0
]~, [1, -24, 0]~, [1, 0, -24]~], [Vecsmall([]), Vecsmall([]), Vecsmall([])],
 [1/3, 0, 0; 0, 1/3, 0; 0, 0, 1/3]], [[3, 3, 3], [[10, 0, 0]~, [1, 9, 0]~, [
1, 0, 9]~], [[-17, 0, 0]~, [1, -18, 0]~, [1, 0, -18]~], [Vecsmall([]), Vecsm
all([]), Vecsmall([])], [1/9, 0, 0; 0, 1/9, 0; 0, 0, 1/9]]], [[], [], []]], 
[468, -77, 728, 0, -1456, 546, 0, -1092; 0, 0, 0, 1, 0, 0, -6, 0; 0, 1, -1, 
0, 1, -3, 0, 3]]
? ideallog(nf2,y,bid)
[750, 3, 8]~
? idealmin(nf,idx,[1,2,3])
[1, 0, 1, 0, 0]~
? idealnorm(nf,idt)
16
? idp=idealpow(nf,idx,7)

[2187 1436 1807 630 1822]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idealpow(nf,idx,7,1)

[1 0 0 0 0]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

? idealprimedec(nf,2)
[[2, [3, 0, 1, 0, 0]~, 1, 1, [0, 0, 0, 1, 1]~], [2, [12, -4, -2, 11, 3]~, 1,
 4, [1, 0, 1, 0, 0]~]]
? idealprimedec(nf,3)
[[3, [1, 0, 1, 0, 0]~, 1, 1, [1, -1, -1, -1, 0]~], [3, [1, 1, 1, 0, 0]~, 2, 
2, [0, 2, 2, 1, 0]~]]
? idealprimedec(nf,11)
[[11, [11, 0, 0, 0, 0]~, 1, 5, 1]]
? idealtwoelt(nf,idy)
[5, [2, 2, 1, 2, 1]~]
? idealtwoelt(nf,idy,10)
[-1, 4, 2, 4, 2]~
? idealstar(nf2,54)
[[[54, 0, 0; 0, 54, 0; 0, 0, 54], [0]], [132678, [1638, 9, 9]], [[2, [2, 0, 
0]~, 1, 3, 1], 1; [3, [3, 0, 0]~, 1, 3, 1], 3], [[[[7], [[1, 0, 1]~], [[1, 0
, -27]~], [Vecsmall([])], 1]], [[[26], [[4, 2, 2]~], [[-23, 2, 2]~], [Vecsma
ll([])], 1], [[3, 3, 3], [[4, 0, 0]~, [1, 3, 0]~, [1, 0, 3]~], [[-23, 0, 0]~
, [1, -24, 0]~, [1, 0, -24]~], [Vecsmall([]), Vecsmall([]), Vecsmall([])], [
1/3, 0, 0; 0, 1/3, 0; 0, 0, 1/3]], [[3, 3, 3], [[10, 0, 0]~, [1, 9, 0]~, [1,
 0, 9]~], [[-17, 0, 0]~, [1, -18, 0]~, [1, 0, -18]~], [Vecsmall([]), Vecsmal
l([]), Vecsmall([])], [1/9, 0, 0; 0, 1/9, 0; 0, 0, 1/9]]], [[], [], []]], [4
68, 469, 0, 0, -364, 0, 0, -1092; 0, 0, 1, 0, -6, -6, 0, 0; 0, 0, 0, 1, -3, 
0, -6, 0]]
? idealval(nf,idp,vp)
7
? ba=nfalgtobasis(nf,x^3+5)
[6, 1, 3, 1, 3]~
? bb=nfalgtobasis(nf,x^3+x)
[1, 1, 4, 1, 3]~
? bc=matalgtobasis(nf,[x^2+x;x^2+1])

[[3, -2, 1, 1, 0]~]

[[4, -2, 0, 1, 0]~]

? matbasistoalg(nf,bc)

[Mod(x^2 + x, x^5 - 5*x^3 + 5*x + 25)]

[Mod(x^2 + 1, x^5 - 5*x^3 + 5*x + 25)]

? nfbasis(x^3+4*x+5)
[1, x, 1/7*x^2 - 1/7*x - 2/7]
? nfbasis(x^3+4*x+5,2)
[1, x, 1/7*x^2 - 1/7*x - 2/7]
? nfbasis(x^3+4*x+12,1)
[1, x, 1/2*x^2]
? nfbasistoalg(nf,ba)
Mod(x^3 + 5, x^5 - 5*x^3 + 5*x + 25)
? nfbasis(p2,0,fa)
[1, x, x^2, 1/11699*x^3 + 1847/11699*x^2 - 132/11699*x - 2641/11699, 1/13962
3738889203638909659*x^4 - 1552451622081122020/139623738889203638909659*x^3 +
 418509858130821123141/139623738889203638909659*x^2 - 6810913798507599407313
4/139623738889203638909659*x - 13185339461968406/58346808996920447]
? nfdisc(x^3+4*x+12)
-1036
? nfdisc(x^3+4*x+12,1)
-1036
? nfdisc(p2,0,fa)
136866601
? nfeltdiv(nf,ba,bb)
[584/373, 66/373, -32/373, -105/373, 120/373]~
? nfeltdiveuc(nf,ba,bb)
[2, 0, 0, 0, 0]~
? nfeltdivrem(nf,ba,bb)
[[2, 0, 0, 0, 0]~, [4, -1, -5, -1, -3]~]
? nfeltmod(nf,ba,bb)
[4, -1, -5, -1, -3]~
? nfeltmul(nf,ba,bb)
[50, -15, -35, 60, 15]~
? nfeltpow(nf,bb,5)
[-291920, 136855, 230560, -178520, 74190]~
? nfeltreduce(nf,ba,idx)
[1, 0, 0, 0, 0]~
? nfeltval(nf,ba,vp)
0
? nffactor(nf2,x^3+x)

[x 1]

[x^2 + 1 1]

? aut=nfgaloisconj(nf3)
[-x, x, -1/12*x^4 - 1/2*x, -1/12*x^4 + 1/2*x, 1/12*x^4 - 1/2*x, 1/12*x^4 + 1
/2*x]~
? nfgaloisapply(nf3,aut[5],Mod(x^5,x^6+108))
Mod(-1/2*x^5 + 9*x^2, x^6 + 108)
? nfhilbert(nf,3,5)
-1
? nfhilbert(nf,3,5,vp)
-1
? nfhnf(nf,[a,aid])
[[1, 1, 4; 0, 1, 0; 0, 0, 1], [[15, 2, 10, 12, 4; 0, 1, 0, 0, 0; 0, 0, 1, 0,
 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0
, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 
0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]]]
? da=nfdetint(nf,[a,aid])

[15 10 5 0 12]

[0 5 0 0 2]

[0 0 5 0 1]

[0 0 0 5 2]

[0 0 0 0 1]

? nfhnfmod(nf,[a,aid],da)
[[1, 1, 4; 0, 1, 0; 0, 0, 1], [[15, 2, 10, 12, 4; 0, 1, 0, 0, 0; 0, 0, 1, 0,
 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 0
, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0, 0; 0, 0, 1, 
0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]]]
? nfisideal(bnf[7],[5,1;0,1])
1
? nfisincl(x^2+1,x^4+1)
[-x^2, x^2]
? nfisincl(x^2+1,nfinit(x^4+1))
[-x^2, x^2]
? nfisisom(x^3+x^2-2*x-1,x^3+x^2-2*x-1)
[x, -x^2 - x + 1, x^2 - 2]
? nfisisom(x^3-2,nfinit(x^3-6*x^2-6*x-30))
[-1/25*x^2 + 13/25*x - 2/5]
? nfroots(nf2,x+2)
[Mod(-2, y^3 - y - 1)]
? nfrootsof1(nf)
[2, -1]
? nfsnf(nf,[as,[1,1,1],[idealinv(nf,idx),idealinv(nf,idy),1]])
[[15706993357777254170417850, 1636878763571210697462070, 1307908830618593502
9427775, 1815705333955314515809980, 7581330311082212790621785; 0, 5, 0, 0, 0
; 0, 0, 5, 0, 0; 0, 0, 0, 5, 0; 0, 0, 0, 0, 5], [1, 0, 0, 0, 0; 0, 1, 0, 0, 
0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1], [1, 0, 0, 0, 0; 0, 1, 0, 0,
 0; 0, 0, 1, 0, 0; 0, 0, 0, 1, 0; 0, 0, 0, 0, 1]]
? nfsubfields(nf)
[[x, 0], [x^5 - 5*x^3 + 5*x + 25, x]]
? polcompositum(x^4-4*x+2,x^3-x-1)
[x^12 - 4*x^10 + 8*x^9 + 12*x^8 + 12*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*x
^2 - 128*x - 5]
? polcompositum(x^4-4*x+2,x^3-x-1,1)
[[x^12 - 4*x^10 + 8*x^9 + 12*x^8 + 12*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*
x^2 - 128*x - 5, Mod(-279140305176/29063006931199*x^11 + 129916611552/290630
06931199*x^10 + 1272919322296/29063006931199*x^9 - 2813750209005/29063006931
199*x^8 - 2859411937992/29063006931199*x^7 - 414533880536/29063006931199*x^6
 - 35713977492936/29063006931199*x^5 - 17432607267590/29063006931199*x^4 + 4
9785595543672/29063006931199*x^3 + 9423768373204/29063006931199*x^2 - 427797
76146743/29063006931199*x + 37962587857138/29063006931199, x^12 - 4*x^10 + 8
*x^9 + 12*x^8 + 12*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*x^2 - 128*x - 5), M
od(-279140305176/29063006931199*x^11 + 129916611552/29063006931199*x^10 + 12
72919322296/29063006931199*x^9 - 2813750209005/29063006931199*x^8 - 28594119
37992/29063006931199*x^7 - 414533880536/29063006931199*x^6 - 35713977492936/
29063006931199*x^5 - 17432607267590/29063006931199*x^4 + 49785595543672/2906
3006931199*x^3 + 9423768373204/29063006931199*x^2 - 13716769215544/290630069
31199*x + 37962587857138/29063006931199, x^12 - 4*x^10 + 8*x^9 + 12*x^8 + 12
*x^7 + 138*x^6 + 132*x^5 - 43*x^4 + 58*x^2 - 128*x - 5), -1]]
? polgalois(x^6-3*x^2-1)
[12, 1, 1, "A_4(6) = [2^2]3"]
? polred(x^5-2*x^4-4*x^3-96*x^2-352*x-568)
[x - 1, x^5 - x^4 - 6*x^3 + 6*x^2 + 13*x - 5, x^5 - x^4 + 2*x^3 - 4*x^2 + x 
- 1, x^5 - x^4 + 4*x^3 - 2*x^2 + x - 1, x^5 + 4*x^3 - 4*x^2 + 8*x - 8]
? polred(x^4-28*x^3-458*x^2+9156*x-25321,3)

[1 x - 1]

[1/115*x^2 - 14/115*x - 327/115 x^2 - 10]

[2/897*x^3 - 14/299*x^2 - 1171/897*x + 9569/897 x^4 - 32*x^2 + 6]

[1/4485*x^3 - 7/1495*x^2 - 1034/4485*x + 7924/4485 x^4 - 8*x^2 + 6]

? polred(x^4+576,1)
[x - 1, x^2 - x + 1, x^2 + 1, x^4 - x^2 + 1]
? polred(x^4+576,3)

[1 x - 1]

[-1/192*x^3 - 1/8*x + 1/2 x^2 - x + 1]

[1/24*x^2 x^2 + 1]

[1/192*x^3 + 1/48*x^2 - 1/8*x x^4 - x^2 + 1]

? polred(p2,0,fa)
[x - 1, x^5 - 2*x^4 - 62*x^3 + 85*x^2 + 818*x + 1, x^5 - 2*x^4 - 53*x^3 - 46
*x^2 + 508*x + 913, x^5 - 2*x^4 - 13*x^3 + 37*x^2 - 21*x - 1, x^5 - x^4 - 52
*x^3 - 197*x^2 - 273*x - 127]
? polred(p2,1,fa)
[x - 1, x^5 - 2*x^4 - 62*x^3 + 85*x^2 + 818*x + 1, x^5 - 2*x^4 - 53*x^3 - 46
*x^2 + 508*x + 913, x^5 - 2*x^4 - 13*x^3 + 37*x^2 - 21*x - 1, x^5 - x^4 - 52
*x^3 - 197*x^2 - 273*x - 127]
? polredabs(x^5-2*x^4-4*x^3-96*x^2-352*x-568)
x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1
? polredabs(x^5-2*x^4-4*x^3-96*x^2-352*x-568,1)
[x^5 - x^4 + 2*x^3 - 4*x^2 + x - 1, Mod(2*x^4 - x^3 + 3*x^2 - 3*x - 1, x^5 -
 x^4 + 2*x^3 - 4*x^2 + x - 1)]
? polredord(x^3-12*x+45*x-1)
[x - 1, x^3 - 363*x - 2663, x^3 + 33*x - 1]
? polsubcyclo(31,5)
x^5 + x^4 - 12*x^3 - 21*x^2 + x + 5
? setrand(1);poltschirnhaus(x^5-x-1)
x^5 + 10*x^4 + 32*x^3 - 100*x^2 - 879*x - 1457
? p=x^5-5*x+y;aa=rnfpseudobasis(nf2,p)
[[1, 0, 0, -2, [3, 1, 0]~; 0, 1, 0, 2, [0, -1, 0]~; 0, 0, 1, 1, [-5, -2, 0]~
; 0, 0, 0, 1, -2; 0, 0, 0, 0, 1], [1, 1, 1, [1, 0, 2/5; 0, 1, 3/5; 0, 0, 1/5
], [1, 0, 22/25; 0, 1, 8/25; 0, 0, 1/25]], [416134375, 202396875, 60056800; 
0, 3125, 2700; 0, 0, 25], [-1275, 5, 5]~]
? rnfbasis(bnf2,aa)

[1 0 0 [-26/25, 11/25, -8/25]~ [0, 4, -7]~]

[0 1 0 [53/25, -8/25, -1/25]~ [6/5, -41/5, 53/5]~]

[0 0 1 [-14/25, -21/25, 13/25]~ [-16/5, 1/5, 7/5]~]

[0 0 0 [7/25, -2/25, 6/25]~ [2/5, -2/5, 11/5]~]

[0 0 0 [9/25, 1/25, -3/25]~ [2/5, -7/5, 6/5]~]

? rnfdisc(nf2,p)
[[416134375, 202396875, 60056800; 0, 3125, 2700; 0, 0, 25], [-1275, 5, 5]~]
? rnfequation(nf2,p)
x^15 - 15*x^11 + 75*x^7 - x^5 - 125*x^3 + 5*x + 1
? rnfequation(nf2,p,1)
[x^15 - 15*x^11 + 75*x^7 - x^5 - 125*x^3 + 5*x + 1, Mod(-x^5 + 5*x, x^15 - 1
5*x^11 + 75*x^7 - x^5 - 125*x^3 + 5*x + 1), 0]
? rnfhnfbasis(bnf2,aa)

[1 0 0 [-6/5, -4/5, 2/5]~ [3/25, -8/25, 24/25]~]

[0 1 0 [6/5, 4/5, -2/5]~ [-9/25, -1/25, 3/25]~]

[0 0 1 [3/5, 2/5, -1/5]~ [-8/25, 13/25, -39/25]~]

[0 0 0 [3/5, 2/5, -1/5]~ [4/25, 6/25, -18/25]~]

[0 0 0 0 [-2/25, -3/25, 9/25]~]

? rnfisfree(bnf2,aa)
1
? rnfsteinitz(nf2,aa)
[[1, 0, 0, [-26/25, 11/25, -8/25]~, [29/125, -2/25, 8/125]~; 0, 1, 0, [53/25
, -8/25, -1/25]~, [-53/125, 7/125, 1/125]~; 0, 0, 1, [-14/25, -21/25, 13/25]
~, [9/125, 19/125, -13/125]~; 0, 0, 0, [7/25, -2/25, 6/25]~, [-9/125, 2/125,
 -6/125]~; 0, 0, 0, [9/25, 1/25, -3/25]~, [-8/125, -1/125, 3/125]~], [1, 1, 
1, 1, [125, 0, 22; 0, 125, 108; 0, 0, 1]], [416134375, 202396875, 60056800; 
0, 3125, 2700; 0, 0, 25], [-1275, 5, 5]~]
? nfz=zetakinit(x^2-2);
? zetak(nfz,-3)
0.091666666666666666666666666666666666667
? zetak(nfz,1.5+3*I)
0.88324345992059326405525724366416928893 - 0.2067536250233895222724230899142
7938848*I
? setrand(1);quadclassunit(1-10^7,,[1,1])
[2416, [1208, 2], [Qfb(277, 55, 9028), Qfb(1700, 1249, 1700)], 1]
? setrand(1);quadclassunit(10^9-3,,[0.5,0.5])
[4, [4], [Qfb(283, 31285, -18771, 0.E-57)], 2800.625251907016076486370621737
0745514]
? sizebyte(%)
288
? getheap
[175, 102906]
? print("Total time spent: ",gettime);
Total time spent: 108
